The inverse problem for the Steiner-Wiener index of trees

For a connected graph $G$ and a set $S\subset V(G)$, the Steiner distance $d_G(S)$ is the minimum number of edges in a connected subgraph of $G$ containing $S$. The Steiner-Wiener $k$ index is defined by ${\mathrm{S}\mathrm{W}}_k(G)={\sum }_{S\subset V(G),\mathrm{\mid }S\mathrm{\mid }=k}{d}_G(S)$. We study the inverse problem for this invariant restricted to trees: for fixed $k$, which positive integers occur as ${\mathrm{S}\mathrm{W}}_k(T)$ for a finite tree $T$? We prove that all sufficiently large positive integers occur as ${\mathrm{S}\mathrm{W}}_k(T)$ for some finite tree $T$ if and only if $k$ is even. For odd $k$, we further show that the set of attainable values has asymptotic density of order $k^{-\delta }(\log k{)}^{-3\mathrm{/}2}$, where $\delta$ is the Erd\H{o}s-Tenenbaum-Ford constant.